Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces (Q1034242)

Let \(H\) be a real separable Hilbert space, let \(\{X_n;n\geq1\}\) be a sequence of i.i.d. \(H\)-valued random variables with zero mean and covariance operator \(\Sigma\), and put \(S_n=X_1+\dots+X_n\). Following a standard approach, the authors compute \(\lim_{\varepsilon\searrow0}\sum_{n\geq1}(\log n)^{b}n^{-3/2}E\{\|S_{n}\|-\sigma\varepsilon\sqrt{n\log n}\}_+\), where \(b>-1\) and \(\sigma^2\) denotes the largest eigenvalue of \(\Sigma\).